Quantum and Classical Query Complexities of Local Search Are Polynomially Related | Algorithmica Skip to main content
Springer Nature Link
Log in

Quantum and Classical Query Complexities of Local Search Are Polynomially Related

  • Published:
Algorithmica Aims and scope Submit manuscript

Abstract

Let f be an integer valued function on a finite set V. We call an undirected graph G(V,E) a neighborhood structure for f. The problem of finding a local minimum for f can be phrased as: for a fixed neighborhood structure G(V,E) find a vertex x∈V such that f(x) is not bigger than any value that f takes on some neighbor of x. The complexity of the algorithm is measured by the number of questions of the form “what is the value of f on x?” We show that the deterministic, randomized and quantum query complexities of the problem are polynomially related. This generalizes earlier results of Aldous (Ann. Probab. 11(2):403–413, [1983]) and Aaronson (SIAM J. Comput. 35(4):804–824, [2006]) and solves the main open problem in Aaronson (SIAM J. Comput. 35(4):804–824, [2006]).

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Subscribe and save

Springer+ Basic
¥17,985 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Price includes VAT (Japan)

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Aaronson, S.: Lower bounds for local search by quantum arguments. SIAM J. Comput. 35(4), 804–824 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  2. Aaronson, S., Shi, Y.: Quantum lower bounds for the collision and the element distinctness problems. J. Assoc. Comput. Mach. 51(4), 595–605 (2004)

    MATH  MathSciNet  Google Scholar 

  3. Aho, A., Hopcroft, J., Ullman, J.: Data Structures and Algorithms. Addison-Wesley, Reading (1983)

    MATH  Google Scholar 

  4. Aldous, D.: Minimization algorithms and random walk on the d-cube. Ann. Probab. 11(2), 403–413 (1983)

    Article  MATH  MathSciNet  Google Scholar 

  5. Ambainis, A.: Quantum lower bounds by quantum arguments. J. Comput. Syst. Sci. 64, 750–767 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  6. Ambainis, A.: Polynomial degree vs. quantum query complexity. J. Comput. Syst. Sci. 72(2), 220–238 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  7. Barnum, H., Saks, M., Szegedy, M.: Quantum decision trees and semidefinite programming. In: Proc. of the 18th IEEE Conference on Computational Complexity, pp. 179–193 (2003)

  8. Beals, R., Buhrman, H., Cleve, R., Mosca, M., de Wolf, R.: Quantum lower bounds by polynomials. J. Assoc. Comput. Mach. 48(4), 778–797 (2001)

    MATH  Google Scholar 

  9. Bennett, C., Bernstein, E., Brassard, G., Vazirani, U.: Strength and weaknesses of quantum computing. SIAM J. Comput. 26(5), 1510–1523 (1997)

    Article  MATH  MathSciNet  Google Scholar 

  10. Buhrman, H., de Wolf, R.: A lower bound for quantum search of an ordered list. Inf. Process. Lett. 70, 205–209 (1999)

    Article  MATH  Google Scholar 

  11. Chen, X., Deng, X.: On the complexity of 2D discrete fixed point problem. In: Proc. of the 33rd International Colloquium on Automata, Languages and Programming, pp. 489–500 (2006)

  12. Deutsch, D., Jozsa, R.: Rapid solution of problems by quantum computation. In: Proc. of the Royal Society A, vol. 439 (1985)

  13. Dürr, C., Heiligman, M., Høyer, P., Mhalla, M.: Quantum query complexity of some graph problems. SIAM J. Comput. 35(6), 1310–1328 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  14. Friedl, K., Ivanyos, G., Santha, M., Verhoeven, Y.: On the black-box complexity of Sperner’s Lemma. In: Proc. of the 15th Fundamentals of Computation Theory. LNCS, vol. 3623, pp. 245–257. Springer, Berlin (2005)

    Chapter  Google Scholar 

  15. Grover, L.: A fast quantum mechanical algorithm for database search. In: Proc. of the 28th ACM Symposium on Theory of Computing, pp. 212–219 (1996)

  16. Høyer, P., Lee, T., Špalek, R.: Negative weights make adversaries stronger. In: Proc. of the 39th ACM Symposium on Theory of Computing, pp. 526–535 (2007)

  17. Høyer, P., Neerbek, J., Shi, Y.: Quantum complexities of ordered searching, sorting, and element distinctness. Algorithmica 34(4), 429–448 (2002)

    Article  MathSciNet  Google Scholar 

  18. Johnson, D., Papadimitriou, C., Yannakakis, M.: How easy is local search. J. Comput. Syst. Sci. 37, 429–448 (1988)

    Article  MathSciNet  Google Scholar 

  19. Laplante, S., Magniez, F.: Lower bounds for randomized and quantum query complexity using Kolmogorov arguments. In: Proc. of the 19th IEEE Conference on Computational Complexity, pp. 294–304 (2004)

  20. Llewellyn, D., Tovey, C.: Dividing and conquering the square. Discrete Appl. Math. 43, 131–153 (1993)

    Article  MATH  MathSciNet  Google Scholar 

  21. Llewellyn, D., Tovey, C., Trick, M.: Local optimization on graphs. Discrete Appl. Math. 23, 157–178 (1989)

    Article  MATH  MathSciNet  Google Scholar 

  22. Llewellyn, D., Tovey, C., Trick, M.: Local optimization on graphs. Erratum 46, 93–94 (1993)

    MATH  MathSciNet  Google Scholar 

  23. Megiddo, N., Papadimitriou, C.: On total functions, existence theorems, and computational complexity. Theor. Comp. Sci. 81, 317–324 (1991)

    Article  MATH  MathSciNet  Google Scholar 

  24. Menger, K.: Zur allgemeinen Kurventhoerie. Fundam. Math. 10, 96–115 (1927)

    MATH  Google Scholar 

  25. Simon, D.: On the power of quantum computation. SIAM J. Comput. 26(5), 1474–1783 (1997)

    Article  MATH  MathSciNet  Google Scholar 

  26. Špalek, R.: Quantum algorithms, lower bounds, and time-space tradeoffs. Ph.D. Thesis, ILLC Dissertation Series DS-2006-04, Universiteit van Amsterdam (2006)

  27. Špalek, R., Szegedy, M.: All quantum adversary methods are equivalents. Theory Comput. 2, 1–18 (2006)

    MathSciNet  Google Scholar 

  28. Sun, X., Yao, A.: On the quantum query complexity of local search in two and three dimensions. In: Proc. of the 47th IEEE Symposium on Foundations of Computer Science, pp. 429–438 (2006)

  29. Verhoeven, Y.: Enhanced algorithms for local search. Inf. Process. Lett. 97, 171–176 (2006)

    Article  MathSciNet  Google Scholar 

  30. Zalka, C.: Grover’s quantum searching algorithm is optimal. Phys. Rev. A 60, 2746–2751 (1999)

    Article  Google Scholar 

  31. Zhang, S.: On the power of Ambainis’s lower bounds. Theor. Comp. Sci. 339, 241–256 (2005)

    Article  MATH  Google Scholar 

  32. Zhang, S.: New upper and lower bounds for randomized and quantum local search. In: Proc. of the 38th ACM Symposium on Theory of Computing, pp. 634–643 (2006)

Download references

Author information

Authors and Affiliations

  1. LRI, Université Paris-Sud, CNRS, 91405, Orsay, France

    Miklos Santha

  2. Rutgers University, Piscataway, NJ, 08854-8019, USA

    Mario Szegedy

Authors
  1. Miklos Santha
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Mario Szegedy
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Miklos Santha.

Additional information

Research of M. Santha was supported by the European Commission IST projects RESQ 37559 and QAP 015848, and by the ANR Blanc AlgoQP grant of the French Research Ministry.

Research of M. Szegedy was supported by NSF grant 0105692 and the European Commission IST project RESQ 37559. The research was done while the author was visiting LRI, Université Paris–Sud, CNRS.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Santha, M., Szegedy, M. Quantum and Classical Query Complexities of Local Search Are Polynomially Related. Algorithmica 55, 557–575 (2009). https://doi.org/10.1007/s00453-008-9169-z

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00453-008-9169-z

Keywords

Search

Navigation